Obesity Is Defined As Having Body Mass Index (BMI) Of 30 Or

Obesity is defined as having body mass index (BMI) of 30 or higher. There are 3 categories of obesity; Category 1 = BMI 30 to less than 35; Category 2 = BMI 35 to less than 40; and Category 3 = BMI 30 or more. The National Center for Health Statistics reports that the prevalence of different categories of obesity are: Category 1: 60%, category 2: 30%, and category 3: 10%.

Dear Students, please read module 4, Hypothesis Testing for Categorical and Ordinal Outcomes, as well as textbook chapter 7, section 7.4. Carefully review the examples provided to understand the procedures of hypothesis testing. Once confident in your understanding, proceed with the assigned task. If you have any questions, let me know in advance. Thank you.

Assignment

Obesity is classified into three categories based on BMI: Category 1 (BMI 30 to less than 35), Category 2 (BMI 35 to less than 40), and Category 3 (BMI 40 or more). According to the National Center for Health Statistics, the typical distribution of these categories in the population is: 60% in Category 1, 30% in Category 2, and 10% in Category 3. A health, education, and nutrition program was launched over five years to address obesity issues. A survey was conducted post-intervention to assess changes in the distribution of obesity categories.

The survey reports the distribution of individuals across the three categories after the program. Your task is to determine whether there has been a significant shift in the distribution of obesity categories following the intervention, indicating an effective program.

Steps for Hypothesis Testing

1. Set Up Hypotheses and Determine Level of Significance

The null hypothesis (H₀) posits that there is no change in the distribution of obesity categories; the population proportions align with the original distribution: 60%, 30%, and 10%. The alternative hypothesis (H_a) suggests that the distribution has shifted due to the intervention.

Level of significance (α) is set at 0.05, indicating a 5% risk of rejecting the null hypothesis when it is true.

2. Select the Appropriate Test Statistic

The data involves categorical variables with observed frequencies compared to expected proportions. The suitable test here is the Chi-square goodness-of-fit test, which assesses whether the observed distribution significantly deviates from the expected distribution.

3. Set Up the Decision Rule

Calculate the Chi-square statistic based on observed and expected frequencies. Compare this value to the critical value from the Chi-square distribution table with degrees of freedom equal to number of categories minus 1 (df=2 for three categories). If the computed Chi-square exceeds the critical value at α=0.05, reject H₀.

4. Compute the Test Statistic

Assuming the sample data gives observed frequencies for each category, compute expected frequencies by multiplying total sample size by the expected proportions (0.60, 0.30, 0.10). Then, apply the Chi-square formula:

χ² = Σ (O_i - E_i)² / E_i

where O_i are observed frequencies and E_i are expected frequencies.

5. Draw Conclusions

Based on the computed Chi-square statistic and the critical value, determine whether to accept or reject the null hypothesis. A rejection suggests a significant shift in the distribution of obesity categories post-intervention, indicating the program’s impact.

Analysis Example with Hypothetical Data

Suppose the survey sample included 300 individuals with the following observed counts:

• Category 1: 150
• Category 2: 90
• Category 3: 60

Expected frequencies based on the original proportions:

• Category 1: 300 × 0.60 = 180
• Category 2: 300 × 0.30 = 90
• Category 3: 300 × 0.10 = 30

Calculating Chi-square:

χ² = (150-180)²/180 + (90-90)²/90 + (60-30)²/30 = 900/180 + 0 + 900/30 = 5 + 0 + 30 = 35

Critical value from Chi-square table with df=2 at α=0.05 is approximately 5.991. Since 35 > 5.991, reject H₀, indicating a significant change in the distribution post-program.

Conclusion

If actual survey data yields a Chi-square statistic exceeding the critical value, it provides statistical evidence that the distribution of obesity categories has shifted significantly, likely due to the health and nutrition program. This suggests their effectiveness in altering obesity patterns.

References

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